We employ the Efficient BB84 WCP Decoy State protocol with 3 intensities \(\mu _1\) , \(\mu _2\)  and \(\mu _3\)  that are randomly and independently chosen [1]} In other literature, this is also known as the 2-decoy state protocol.. Alice prepares WCPs with polarization chosen from one of two bases: \(X\in \lbrace H,V\rbrace \)  and \(Z\in \lbrace D, A\rbrace \) . With probability \(P^A_X\)  Alice chooses the \(X\)  basis, and with probability \(P^A_Z=1-P^A_X\)  she chooses the \(Z\)  basis, In efficient BB84 the two bases need not be chosen with equal probability. Instead, the probability is found by optimizing the protocol so as to maximize the secure key length. With the basis chosen, Alice randomly and uniformly encodes bit values 0 or 1 onto the appropriate polarization state, where: \(V,D\equiv 0\)  and \(H,A\equiv 1\) , before choosing the intensity of each pulse: \(k\in \lbrace \mu _1,\mu _2,\mu _3\rbrace \)  with probability \(p_k\) . The intensity values can be any values that satisfy the relations: (i) \(\mu _1>\mu _2>\mu _3\)  and (ii) \(\mu _1>\mu _2+\mu _3\) . Bob randomly chooses to measure the polarization in either the basis \(X\)  or \(Z\) , with probability \(P^B_X\)  and \(P^B_Z=1-P^B_X\) , respectively.
